138 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



marked W, Ijdng below the release pull curve. This energy may be 

 termed the magnetic drag: it is restored to the magnetic field and dis- 

 sipated in the eddy current paths or in other circuits linking the field. 

 Experimental studies have shown that the kinetic energy T of the arma- 

 ture in backstop impact may vary from 20 per cent to 90 per cent of V. 

 Such a wide variation in armature energy has a proportional effect on 

 rebound amplitude. Within certain limitations, the following analysis 

 establishes the relations controlling the ratio W/V, and also serves for 

 the evaluation of the release motion time. 



EQUATIONS APPLYING 



As the motion starts after an initial period of field decay, the relations 

 applying are given approximately by (2) and (3), with (R and F given by 

 (4A) and (5A), and G = Ge ■ The use of (4A) and (5A) is justified by the 

 fact that the lower portion of the demagnetization curve, which applies 

 to the later stages of flux decay, is approximately linear. The assumption 

 that (2) applies, withG = Ge , is justified by the fact, discussed in Section 

 1, that the initial rapid decay of the field in the outer layers of the core 

 is followed by an exponential decay of the greater part (about 70 per 

 cent) of the initial field. Thus if ts is written for 47r(T£/(R(0) in (2), this 

 expression applies in the latter stages of flux decaj^ when the armature 

 is at rest at x = 0. In this case, the value of Ie is a constant, even if the 

 values of Gb and (R(0) applying differ somewhat from those appljdng 

 in the operate case. 



Taking these equations to appl}^ it is desired to determine the solution 

 for the initial conditions applying to the release case. This cannot be 

 obtained by the approximations used for the operate case, as the sim- 

 plifying condition of a nearly constant reluctance does not apply: the 

 change of reluctance with x is a maximum for x = 0. The procedure 

 that has been employed is the use of the analogue computer described 

 in articles bj' A. A. Currie and E. Lakatos to obtain solutions for a 

 restricted category of cases. The equations may be reduced to a form 

 adapted to such solution as follows: 



An expression for (R/6^(0) obtained from (4A) is substituted in (2). 

 Writing ts for 47rG^£/(R(0), there is obtained: 



Cl{1 -hu) , Ie d^p' „ 

 C,u ^ + 2" -^ = ^' 



where ii is written for x/xo , or .r (.4(Ro). B}^ substituting in (3) the ex- 

 pression for F given by (5A) there is obtained an expression for <p . 



